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Theorem reusv3 4219
Description: Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 4217 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)
Hypotheses
Ref Expression
reusv3.1 (𝑦 = 𝑧 → (𝜑𝜓))
reusv3.2 (𝑦 = 𝑧𝐶 = 𝐷)
Assertion
Ref Expression
reusv3 (∃𝑦𝐵 (𝜑𝐶𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐶,𝑧   𝑥,𝐷,𝑦   𝜑,𝑥,𝑧   𝜓,𝑥,𝑦   𝑥,𝐴,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑧)   𝐶(𝑦)   𝐷(𝑧)

Proof of Theorem reusv3
StepHypRef Expression
1 reusv3.1 . . . . 5 (𝑦 = 𝑧 → (𝜑𝜓))
2 reusv3.2 . . . . . 6 (𝑦 = 𝑧𝐶 = 𝐷)
32eleq1d 2122 . . . . 5 (𝑦 = 𝑧 → (𝐶𝐴𝐷𝐴))
41, 3anbi12d 450 . . . 4 (𝑦 = 𝑧 → ((𝜑𝐶𝐴) ↔ (𝜓𝐷𝐴)))
54cbvrexv 2551 . . 3 (∃𝑦𝐵 (𝜑𝐶𝐴) ↔ ∃𝑧𝐵 (𝜓𝐷𝐴))
6 nfra2xy 2381 . . . . 5 𝑧𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷)
7 nfv 1437 . . . . 5 𝑧𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)
86, 7nfim 1480 . . . 4 𝑧(∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
9 risset 2369 . . . . . 6 (𝐷𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝐷)
10 ralcom 2490 . . . . . . . . . . . . . 14 (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧𝐵𝑦𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
11 impexp 254 . . . . . . . . . . . . . . . . . 18 (((𝜑𝜓) → 𝐶 = 𝐷) ↔ (𝜑 → (𝜓𝐶 = 𝐷)))
12 bi2.04 241 . . . . . . . . . . . . . . . . . 18 ((𝜑 → (𝜓𝐶 = 𝐷)) ↔ (𝜓 → (𝜑𝐶 = 𝐷)))
1311, 12bitri 177 . . . . . . . . . . . . . . . . 17 (((𝜑𝜓) → 𝐶 = 𝐷) ↔ (𝜓 → (𝜑𝐶 = 𝐷)))
1413ralbii 2347 . . . . . . . . . . . . . . . 16 (∀𝑦𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∀𝑦𝐵 (𝜓 → (𝜑𝐶 = 𝐷)))
15 r19.21v 2413 . . . . . . . . . . . . . . . 16 (∀𝑦𝐵 (𝜓 → (𝜑𝐶 = 𝐷)) ↔ (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
1614, 15bitri 177 . . . . . . . . . . . . . . 15 (∀𝑦𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
1716ralbii 2347 . . . . . . . . . . . . . 14 (∀𝑧𝐵𝑦𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧𝐵 (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
1810, 17bitri 177 . . . . . . . . . . . . 13 (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧𝐵 (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
19 rsp 2386 . . . . . . . . . . . . 13 (∀𝑧𝐵 (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷)) → (𝑧𝐵 → (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷))))
2018, 19sylbi 118 . . . . . . . . . . . 12 (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → (𝑧𝐵 → (𝜓 → ∀𝑦𝐵 (𝜑𝐶 = 𝐷))))
2120com3l 79 . . . . . . . . . . 11 (𝑧𝐵 → (𝜓 → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∀𝑦𝐵 (𝜑𝐶 = 𝐷))))
2221imp31 247 . . . . . . . . . 10 (((𝑧𝐵𝜓) ∧ ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷)) → ∀𝑦𝐵 (𝜑𝐶 = 𝐷))
23 eqeq1 2062 . . . . . . . . . . . . 13 (𝑥 = 𝐷 → (𝑥 = 𝐶𝐷 = 𝐶))
24 eqcom 2058 . . . . . . . . . . . . 13 (𝐷 = 𝐶𝐶 = 𝐷)
2523, 24syl6bb 189 . . . . . . . . . . . 12 (𝑥 = 𝐷 → (𝑥 = 𝐶𝐶 = 𝐷))
2625imbi2d 223 . . . . . . . . . . 11 (𝑥 = 𝐷 → ((𝜑𝑥 = 𝐶) ↔ (𝜑𝐶 = 𝐷)))
2726ralbidv 2343 . . . . . . . . . 10 (𝑥 = 𝐷 → (∀𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∀𝑦𝐵 (𝜑𝐶 = 𝐷)))
2822, 27syl5ibrcom 150 . . . . . . . . 9 (((𝑧𝐵𝜓) ∧ ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷)) → (𝑥 = 𝐷 → ∀𝑦𝐵 (𝜑𝑥 = 𝐶)))
2928reximdv 2437 . . . . . . . 8 (((𝑧𝐵𝜓) ∧ ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷)) → (∃𝑥𝐴 𝑥 = 𝐷 → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
3029ex 112 . . . . . . 7 ((𝑧𝐵𝜓) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → (∃𝑥𝐴 𝑥 = 𝐷 → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))))
3130com23 76 . . . . . 6 ((𝑧𝐵𝜓) → (∃𝑥𝐴 𝑥 = 𝐷 → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))))
329, 31syl5bi 145 . . . . 5 ((𝑧𝐵𝜓) → (𝐷𝐴 → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))))
3332expimpd 349 . . . 4 (𝑧𝐵 → ((𝜓𝐷𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))))
348, 33rexlimi 2443 . . 3 (∃𝑧𝐵 (𝜓𝐷𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
355, 34sylbi 118 . 2 (∃𝑦𝐵 (𝜑𝐶𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) → ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
361, 2reusv3i 4218 . 2 (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
3735, 36impbid1 134 1 (∃𝑦𝐵 (𝜑𝐶𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329
This theorem is referenced by: (None)
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